Asymptotic structures of cardinals (Q2922051)

A ballean is a triple \((X,P,B)\), where \(X\) and \(P\) are non-empty sets and \(B:X\times P\to 2^X\) such that \(x\in B(x,\alpha)\), for any \(\alpha\in P\) there are \(\alpha',\alpha''\in P\) such that \(B(x,\alpha)\subset B^*(x,\alpha')=\{y\in X\mid x\in B(y,\alpha')\}\) and \(B^*(x,\alpha)\subset B(x,\alpha'')\), for any \(\alpha,\beta\in P\) there is \(\gamma\in P\) such that \(B(B(x,\alpha),\beta)\subset B(x,\gamma)\) for each \(x\in X\) and for each \(x,y\in X\) there is \(\alpha\in P\) with \(y\in B(x,\alpha)\). The ballean \((\kappa,\kappa, \overset {B}\leftrightarrow)\), for a given cardinal \(\kappa\), where \(\overset{B}{\leftrightarrow}(x,\alpha)=\{y\in\kappa\mid x\in[y,y+\alpha] \text{ or } y\in[x,x+\alpha]\}\) is the main object of investigation, particularly when it is equivalent to a metric ballean and some relevant cardinal invariants.